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12 Basic Mathematical Relations

Submitted by DB Larson on Mon, 08/04/2008 - 12:36

CHAPTER 12

Basic Mathematical Relations

It was pointed out in the introductory chapters that when we postulate a universe composed entirely of motion, every entity or phenomenon that exists in this universe is either a motion, a combination of motions, or a relation between motions. The discussion thus far has been addressed mainly to an examination of the primary features of the possible motions, and certain of the combinations of these motions. At this point it will be advisable to consider some of the basic kinds of relations that exist between motions.

Inasmuch as motion in general is defined as a relation between space and time, expressed symbolically by s/t, all of the different kinds of motions, and the relations between motions, can be expressed in space-time terms. Such an analysis into space and time components will be particularly helpful in putting the various physical relationships into the proper perspective, and our first objective in the field we are now entering will therefore be to establish the space-time equivalents of the various quantities that constitute the so-called “mechanical” system. Consideration of the analogous quantities of the electrical system will be deferred until we are ready to begin an examination of electrical phenomena.

One set of these mechanical quantities is customarily expressed in velocity terms, and it presents no problems. One-dimensional velocity is, by definition, s/t. It follows that two-dimensional and three-dimensional velocity is s2/t2 and s3/t3 respectively. Acceleration, the time rate of change of one-dimensional velocity, is s/t2.

In addition to these quantities which express motion as velocities (or speeds), there is also a set of quantities which are fundamentally based on resistance to movement, although in some applications this basic significance is obscured by other factors. The objects, which resist movement, are atoms and particles of matter: three-dimensional combinations of motions. In a universe of motion, where nothing exists but motion, the only thing that can resist change of motion is motion. The particular motion that resists any change in the motion of an atom is the inherent motion of the atom itself, the motion that makes it an atom. Furthermore, only a three-dimensional motion, or motion that is automatically distributed over three dimensions, is able to offer effective resistance, as any vacant dimension permits motion to take place without hindrance.

The magnitude of the resistance can be expressed in terms of the quantity required to eliminate the effective existing motion; that is, to reduce this motion to unity in the conventional reference system. This is the inverse of the motion of the atom, s3/t3, and the resistance to motion, or inertia, is therefore t3/s3. In more general application, inertia is known as mass.

Inasmuch as current physical theory recognizes gravitation and inertia a as phenomena of a quite different character, the equivalence of gravitational and inertial mass, which has been experimentally demonstrated to the almost incredible accuracy of less than one part in 1011, is regarded as very significant, although there is considerable difference of opinion as to what that significance actually is. As expressed by Clifford M. Will, “the theoretical interpretation of the Eötvös experiment (which demonstrates the equivalence) has varied.”57 Will asserts that it is now believed that the results of this experiment rule out all non-metric theories of gravitation (he defines metric theories as those “in which gravitation can be treated as being synonymous with the curvature of space and time” ). After the theorists have arrived at such a far-reaching conclusion on the basis of what Will admits is no more than a “conjecture,” it comes as something of an anticlimax when the Reciprocal System reveals that nothing of an esoteric nature is involved. Gravitation is a motion, but it can manifest itself either directly as motion or inversely as resistance to another motion.

Multiplying mass, t3/s3, by velocity, s/t, we obtain momentum, t2/s2, the reciprocal of two-dimensional velocity. Another multiplication by velocity, s/t, gives us energy, t/s. Energy, then, is the reciprocal of velocity. When one-dimensional motion is not restrained by opposing motion (force) it manifests itself as velocity; when it is so restrained it manifests itself as potential energy. Kinetic energy is merely “energy in transit,” so to speak. It is a measure of the energy that has been used to produce the velocity of a mass (½ mv2 = ½ t3/s3 × s2/t2 = ½ t/s), and can be extracted for other use by terminating the motion (velocity).

This explanation of the nature of energy should be of some assistance to those who are still having some difficulty with the concept of scalar motion. Both speed and energy are scalar measures of motion. But on our side of the unit speed boundary, the low-speed side, where all motion is in space, speed can be represented in our conventional spatial system of reference because it causes a change of position, inward or outward, in space, whereas energy cannot be so represented. On the high speed side of the boundary, the relations are inverted. There all motion is in time, and the measure of that motion, the energy, t/s, the inverse of speed, s/t, can be represented in a stationary temporal reference system, whereas speed is neither inward nor outward from the time standpoint, and cannot be represented in the temporal coordinate system.

Here is the reason for the purely scalar nature of any increment of speed beyond the unit level, such as those discussed in Chapter 8. The added speed does have a direction, but it is a direction in time, and it has no vectorial effect in a spatial system of reference. We will find this very significant when we undertake an examination of some of the recently discovered high speed astronomical objects in Volume II.

Force, which is defined as the product of mass and acceleration, becomes t3/s3 × s/t2 = t/s2. Acceleration and force are thus inverse quantities, in the sense in which that term is generally used in this work; that is, they are identical except that space and time are interchanged. They are not inverse in the mathematical sense, as their product is not equal to unity.

One special type of force that is of particular interest is the gravitational force, that which the aggregates of matter appear to exert on each other by reason of there motions inward in space. In this case, the mathematical expression F = kmm’/d2 by which the force is ordinarily calculated is quite different from the general force equation F= ma. When taken at their face value, these two expressions are clearly irreconcilable. If gravitational force is actually a force, even a force of the “as if,” variety, it cannot be proportional to the product of two masses (that is, to m2) when force in general is proportional to the first power of the mass. There is an obvious contradiction here.

Most of the other common quantities of the mechanical system can be reduced to space-time terms without any complications. For example:

Impulse, the product of force and time, has the same dimensions as momentum.

Ft = t/s2 × t = t2/s2

Both work and torque are the products of force and distance, and have the same dimensions as energy.

Fs = t/s2 × s = t/s

Pressure is force per unit area.

F/s2 = t/s × 1/s2 = t/s4

Density is mass per unit volume.

m/s3 = t3/s3 × 1/s3 = t3/s6

Viscosity is mass per unit length per unit time.

m × 1/s × 1/t = t3/s3 × 1/s × 1/t = t2/s4

Surface tension is force per unit length.

F/s = t/s2 × 1/s = t/s3

Power is work per unit time.

W/t = t/s × 1/t = 1/s

All of the established relations in the field of mechanics have the same dimensional consistency on the basis of these space-time dimensions as in the conventional forms, since the mass terms in the equations are, in all cases, balanced by derivatives of mass on the opposite side of the equation. The numerical values in these equations likewise retain the same relationships, as all that we have done, from this standpoint, is to change the size of the unit in which the quantity of mass is expressed. What has been accomplished, then, is to express mass in terms of the components of motion. Since mechanics deals only with space, time, and mass, it follows that, so far as mechanics is concerned, by reducing mass to motion we have confirmed the validity of the basic postulate that the physical universe is composed entirely of motion.

This is a very significant point. The concept of the nature of the physical universe on which conventional physics is based, the concept of a universe of matter existing in a framework provided by space and time, identifies matter as a fundamental quantity. The results of this present work now show that, in the physical field that is the most completely developed and understood, the fundamental entity is motion, not matter. Furthermore, it is now possible to see why the common denominator of the universe has to be motion; why it could not be anything else. It has to be something to which all of the mechanical quantities can be reduced (and all other physical quantities as well, but for the present we are examining the mechanical relations). The only entity that meets these requirements is the simple relationship between space and time that we are defining as motion. Motion is the common denominator of the field of mechanics.

It still remains to be established that motion is the common denominator of the entire universe, but the demonstration that all of the quantities with which mechanics deals, including mass , can be reduced to motion creates a strong presumption that when the more complex phenomena in other fields are equally well understood they will also be found to be reducible to motion. The development of theory in the subsequent pages of this and the volumes to follow will show that this logical expectation is realized, and that all physical phenomena and entities can, in fact, be reduced to motion.

The application of the Reciprocal System of theory to mechanics throws a significant light on the relation of this theoretical system to conventional scientific thought. It was asserted in Chapter 6 that the concept of a universe of motion, on which the new theoretical system is based, is “just the kind of a conceptual alteration that is needed to clear up the existing physical situation: one which makes drastic changes where such changes are required, but leaves the empirically determined relations of our everyday experience essentially untouched.” Here, in application to a field in which the entire body of knowledge is a network of “empirically determined relations,” the validity of this assertion is dramatically demonstrated. The only change that is found to be necessary in mechanics is to recognize the fact that mass is reducible to motion. Otherwise, the entire structure of mechanical theory is incorporated into the Reciprocal System just as it stands. As will be shown in the pages that follow, the same is true in other fields to the extent that the prevailing ideas in those fields are, like the principles of mechanics, solidly based on empirically determined facts. But where the prevailing ideas are based on assumptions—“free inventions of the human mind,” in Einstein’s words—the development of the theory of a universe of motion now shows that most of these invented ideas are erroneous, in part if not in their entirety. The Reciprocal System diverges from current scientific thought only in those respects where current theory has been led astray by erroneous assumptions. As indicated earlier, the phenomena involved are mainly those not accessible to direct apprehension, primarily the phenomena of the very small, the very large, and the very fast.

In all of the space-time expressions of physical quantities that were derived in the preceding pages of this chapter, the dimensions of the denominator of the fraction are either equal to or greater than the dimensions of the numerator. This is another result of the discrete unit postulate, which prevents any interactions from being carried beyond the unit level. Addition of speed displacement to motion in space reduces the speeds; the atomic rotation can take place only in the negative scalar direction, and so on. The same principle applies to the dimensions of physical quantities, and the dimensions of the numerator of the space-time expression of any real physical quantity cannot be greater than those of the denominator. Purely mathematical relations that violate this principle can, of course, be constructed, but according to the theoretical findings they have no real physical significance.

For example, the reciprocal of viscosity is known as fluidity, and in certain applications it is more convenient for purposes of calculation to work with fluidity values rather than viscosity values. But the space-time expression for fluidity is s4/t2, and on the basis of the principle just stated, we must conclude that viscosity is the quantity that has a real physical existence.

The most notable of the quantities excluded by this dimensional principle is “action.” This is the product of energy, t/s, and time t, and in space-time terms it is t2/s. Thus it is not admissible as a real physical quantity. In view of the prominent place which it occupies in some physical areas, this conclusion that it has no actual physical significance may come as quite a surprise, but the explanation can be seen if we examine the most familiar of the conventional applications of action: its use in the expression of Planck’s constant. The equation connecting the energy of radiation with the frequency is

E = hv

where h is Planck’s constant. In order to be dimensionally consistent with the other quantities in the equation this constant must be expressed in terms of action.

It is clear, however, from the explanation of the nature of the photon of radiation that was developed in Chapter 4, that the so-called “frequency” is actually a speed. It can be expressed as a frequency only because the space that is involved is always a unit magnitude. In reality, the space dimension belongs with the frequency, not with the Planck constant. When it is thus transferred, the remaining dimensions of the constant are t2/s2, which are the dimensions of momentum, and are the reversing dimensions that are required to convert speed s/t to energy t/s. In space-time terms, the equation for the energy of radiation is

t/s = t2/s2 × s/t

Similar situations have developed in other cases where dimensions have been improperly assigned in current practice. The energy of rotation, for instance, is commonly expressed as ½ Iw 2, where I is the moment of inertia, and w is the angular velocity. The moment of inertia is the product of the mass and the square of the distance: I = ms2 = t3/s3 × s2 = t3/s

This result shows that the moment of inertia is an artificial construct without physical significance. The important part that it plays in the expression for rotational energy may seem inconsistent with this conclusion, but again the explanation is that the space magnitude has been improperly assigned. It belongs with the velocity term, not with the mass term. When it is so transferred, the moment of inertia is eliminated, and the rotational energy equation reverts to the normal kinetic form E= ½mv2. The equation in its usual form is merely a mathematical convenience, and does not reflect the actual physical situation.

In addition to the kinds of relations that have been discussed so far in this chapter, where the relations themselves are familiar, and only the analysis into space and time components is new, there are other types of physical relations that are peculiar to the universe of motion. At this time we will want to examine two of these: the limitations on unidirectional motion, and the relations between motion in space and motion in time.

The translational and vibrational speeds with which we have been mainly concerned thus far are speeds attained by means of directional reversals, and their magnitudes are not subject to any limits other than those arising from the finite capabilities of the originating processes. Rotation, however, is unidirectional from the scalar standpoint, and unidirectional magnitudes are limited by the discrete unit postulate. On the basis of this postulate, the maximum possible one-dimensional unidirectional speed is one net displacement unit. However, the atom rotates in the inward scalar direction, and inward motion necessarily takes place in opposition to the omnipresent outward motion of the natural reference system. Two inward displacement units are therefore required in order to reach the limit of one net unit. These two units extend from unity in the positive scalar direction (the positive zero, in terms of the natural system) to unity in the negative scalar direction (the negative zero), and they constitute the maximum for any one-dimensional unidirectional motion. In three-dimensional space (or time) there can be two displacement units in each of the three dimensions, and the maximum three-dimensional unidirectional displacement is therefore 23, or 8, units.

There have been some suggestions that the number of possible directions (and consequently displacements) in three-dimensional space ought to be 3 × 2 = 6 rather than 23 = 8. It should therefore be emphasized that we are not dealing with three individual dimensions of motion, we are dealing with three-dimensional motion. The possible directions in a three-dimensional continuum can be visualized by regarding a two-unit cube as being an assemblage of eight one-unit cubes. The diagonals from the center of the assemblage to the opposite corner of each of the cubes then define the eight possible directions.

An important consequence of the fact that there are eight displacement units between the zero point of the positive motion and the end of the second unit, which is the zero from the negative standpoint, is that in any physical situation involving rotation, or other three-dimensional motion, there are eight displacement units between positive and negative magnitudes. A positive displacement x from the positive datum is physically equivalent to a negative displacement 8-x from the negative datum. This is a principle that will have a wide field of application in the pages that follow.

The key factor in the relation between motion in space and motion in time is the previously mentioned fact that in the context of a spatial reference system all motion in time is scalar, and in the context of a temporal reference system all motion in space is scalar. The regions of motion in time and motion in space therefore meet in what is essentially no more than a point contact. It follows that of all of the possible directions that a motion in time can take, only one of these time directions brings the motion in time into contact with the region of motion in space. Only in this one direction can an effect be transmitted across the regional boundary. Inasmuch as all possible directions are equally probable, in the absence of any factors that would establish a preference, the ratio of the transmitted effect to the total magnitude of the motion is numerically equal to the total number of possible directions.

As can be seen from the foregoing explanation, the transmission ratio depends on the nature of the motion, particularly on the number of dimensions involved. However, the value with which we will be most concerned is that applicable to the basic properties of matter. This is the relation that was called the inter-regional ratio in the first edition, and it appears advisable to retain this name, although the more extensive information now available shows the relation is not as general as the name might indicate.

On the basis of the theoretical considerations discussed in the preceding paragraphs, there are 4 possible orientations of each of the two two-dimensional rotations of the atoms, and 8 possible orientations of the one-dimensional rotations, making a total of 4 × 4 × 8 = 128 different positions that a unit displacement of the scalar translational motion of the atom (the inward scalar effect of the rotation) can take in three-dimensional time. In addition, each of the rotating systems of the atom has an initial unit of vibrational displacement with three possible orientations, one in each dimension. For the two-dimensional basic rotation this means nine possible positions, of which two are occupied. Thus, for each of the 128 possible rotational positions there is an additional 2/9 vibrational position which any given displacement unit may occupy. The inter-regional ratio is then 128 (1 + 2/9) = 156.44.

It is this inter-regional ratio that accounts for the small “size” of atoms when the dimensions of these objects are measured on the assumption that they are in contact in the solid state. According to the theory developed in the foregoing pages, there can be no physical distance less than one natural unit, which, as we will see in the next chapter, is 4.56×10-6 cm. But because the inter-atomic equilibrium is established in the region inside this unit, the measured inter-atomic distance is reduced by the inter-regional ratio, and this measured value is therefore in the neighborhood of 10-8 cm.

The inversion of space and time at the unit level also has an important effect on the dimensions of inter-regional relations. Inside unit space no changes in space magnitudes can take place, since less than unit space does not exist. However, as pointed out earlier, the motion in time, which can take place inside the space unit, is equivalent to a motion in space because of the inverse relation between space and time. An increase in the time aspect of a motion in this inside region (the time region, where space remains constant at unity) from 1 to t is equivalent to a decrease in the space aspect from 1 to 1/t. Where the time is t, the speed in this region is equivalent space 1/t divided by time t, or 1/t2.

In the region outside unit space, the speed corresponding to one unit of space and time t is 1/t. Now we find that in the time region it is 1/t2. The time region speed, and all quantities derived therefrom, which means all of the physical phenomena of the inside region, as all of these phenomena are manifestations of motion, are therefore second power expressions of the corresponding quantities of the outside region. This is an important principle that must be taken into account in any relation involving both regions. The intra-region relations may be equivalent; that is, the expression a= be is the mathematical equivalent of the expression a2 = b2c2. But if we measure the quantity a in the outside region, it is essential that the equation be expressed in the correct regional form: a = b2c2.

Although the difficulties which the Reciprocal System of theory does not encounter do not enter into the development of thought in these pages, and, strictly speaking, have no real place in the discussion, it may be of interest, while we are considering some of the factors that enter into the phenomena of very small dimensions, to point out that the theory of a universe of motion is free from the problem of infinities that plagues all conventional theories in this physical area. Richard Feynman gives us a candid assessment of the existing theoretical situation:

We really do not know exactly what it is that we are assuming that gives us the difficulty producing infinities. A nice problem!

However, it turns out that it is possible to sweep the infinities under the rug, by a certain crude skill, and temporarily we are able to keep on calculating…. We have all these nice principles and known facts, but we are in some kind of trouble: either we get the infinities, or we do not get enoughof a description—we are missing some parts.58

The Reciprocal System is free of these problems because it is a fully quantized system of theory. Every physical phenomenon, this theory tells us, is a manifestation of motion, and every motion involves at least one unit of space and one unit of time. For convenience, we may identify a “point” within a unit of space or a unit of time, but such a point has no independent existence. Nothing less than one unit of either space or time exists in the universe of motion.